Sum of first n natural numbers is given by `(n(n+1))/(2)`. What is the geometric mean of the series `1, 2, 4, 8, ……2^(n)?`

Options

A. `2^(n)`
B. `2^((n)/(2))`
C. `2^(1//2)`
D. `2^(n-1)`

Explanation

To find the geometric mean of the series \(1, 2, 4, 8, \ldots, 2^n\), we can follow these steps:

### Step 1: Identify the series
The series given is \(1, 2, 4, 8, \ldots, 2^n\). This can be expressed in terms of powers of 2:
\[ 2^0, 2^1, 2^2, \ldots, 2^n \]

### Step 2: Count the number of terms
The number of terms in this series is \(n + 1\) (from \(2^0\) to \(2^n\)).

### Step 3: Write the formula for the geometric mean
The geometric mean (GM) of a series of numbers \(a_1, a_2, \ldots, a_k\) is given by:
\[ GM = \sqrt[k]{a_1 \cdot a_2 \cdot \ldots \cdot a_k} \]
In our case, \(k = n + 1\).

### Step 4: Calculate the product of the terms
The product of the terms in our series is:
\[ 2^0 \cdot 2^1 \cdot 2^2 \cdots 2^n = 2^{0 + 1 + 2 + \ldots + n} \]
The sum of the first \(n\) natural numbers is given by the formula:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
Thus, we can write:
\[ 2^{0 + 1 + 2 + \ldots + n} = 2^{\frac{n(n + 1)}{2}} \]

### Step 5: Substitute into the GM formula
Now we substitute this product back into the formula for the geometric mean:
\[ GM = \sqrt[n + 1]{2^{\frac{n(n + 1)}{2}}} \]

### Step 6: Simplify the expression
Using the properties of exponents, we can simplify:
\[ GM = 2^{\frac{n(n + 1)}{2(n + 1)}} \]
This simplifies to:
\[ GM = 2^{\frac{n}{2}} \]

### Final Answer
Thus, the geometric mean of the series \(1, 2, 4, 8, \ldots, 2^n\) is:
\[ \boxed{2^{\frac{n}{2}}} \]